Oscillatory neurons and neural networks are found throughout the nervous system. Despite much recent theoretical work on the properties of neural networks, little is known concerning the function of neuronal networks that contain oscillatory elements. The proposed work consists of a series of related modeling nad theoretical investigations aimed at exploring the properties of neural networks with conditionally oscillatory elements. The pyloric network of the crustacean stomatogastric ganglion (STG) will provide the source of biological data that will serve to constrain the models. This preparation is simple enough to allow the formulation of clear hypotheses, yet complex enough so that understanding its properties and modeling it are not trivial. Two kinds of models of individual neurons will be constructed: a) phenomenological "Input-Output" model neurons that mimic the behavior of single neurons of the STG, and b) semi-realistic "Mechanistic Models" that stimulate the results of detailed biophysical data. Networks will be formed from both kinds of model neurons, and analog electronic circuit models constructed as well. These models will be used to study the following questions: a) What different types of firing patterns can the network exhibit? b) How can the behavior of the network be explained on the basis of the properties of the component neurons? c) How do pacemaker and emergent modes of oscillation interact and cooperate to produce stable system behavior? d) How robust is the circuit, and how is it affected by various perturbations? Mathematics relevant to these questions will be developed and explored.